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April  2011, 29(2): 485-503. doi: 10.3934/dcds.2011.29.485

A general theorem on necessary conditions in optimal control

1. 

Université de Lyon, Institut Camille Jordan, CNRS UMR 5208, 69622 Villeurbanne, France

Received  September 2009 Revised  March 2010 Published  October 2010

We give a relatively short and self-contained proof of a theorem that asserts necessary conditions for a general optimal control problem. It has been shown that this theorem, which is simple to state, provides a powerful template from which necessary conditions for various other problems in dynamic optimization can be directly derived, at the level of the state of the art. These include various extensions of the Pontryagin maximum principle and the multiplier rule.
Citation: Francis Clarke. A general theorem on necessary conditions in optimal control. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 485-503. doi: 10.3934/dcds.2011.29.485
References:
[1]

F. H. Clarke, Optimal solutions to differential inclusions, J. Optim. Th. Appl., 19 (1976), 469-478. doi: doi:10.1007/BF00941488.

[2]

F. H. Clarke, "Optimization and Nonsmooth Analysis," Wiley-Interscience, New York, 1983; Republished as vol. 5 of "Classics in Applied Mathematics," SIAM, 1990.

[3]

F. H. Clarke, Necessary conditions in dynamic optimization, Memoirs of the Amer. Math. Soc., 173 (2005), x+113.

[4]

F. H. Clarke, Necessary conditions in optimal control and in the calculus of variations, in "Differential Equations, Chaos and Variational Problems" (ed. V. Staicu), Birkhäuser, (2008), 143-156. doi: doi:10.1007/978-3-7643-8482-1_11.

[5]

F. H. Clarke and M. R. de Pinho, The nonsmooth maximum principle,, Control and Cybernetics, (). 

[6]

F. H. Clarke and M. R. de Pinho, Optimal control problems with mixed constraints,, SIAM J. Control Optim., (). 

[7]

F. H. Clarke, Yu. Ledyaev and M. R. de Pinho, An extension of the Schwarzkopf multiplier rule in optimal control, preprint, 2009.

[8]

F. H. Clarke and Yu. S. Ledyaev, Mean value inequalities in Hilbert space, Trans. Amer. Math. Soc., 344 (1994), 307-324. doi: doi:10.2307/2154718.

[9]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory," Graduate Texts in Mathematics, vol. 178, Springer-Verlag, New York, 1998.

[10]

M. R. Hestenes, "Calculus of Variations and Optimal Control Theory," Wiley, New York, 1966.

[11]

A. A. Milyutin and N. P. Osmolovskii, "Calculus of Variations and Optimal Control," American Math. Soc., Providence, 1998.

[12]

L. W. Neustadt, "Optimization," Princeton University Press, Princeton, 1976.

[13]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mischenko, "The Mathematical Theory of Optimal Processes," Wiley-Interscience, New York, 1962.

[14]

R. B. Vinter, "Optimal Control," Birkhäuser, Boston, 2000.

show all references

References:
[1]

F. H. Clarke, Optimal solutions to differential inclusions, J. Optim. Th. Appl., 19 (1976), 469-478. doi: doi:10.1007/BF00941488.

[2]

F. H. Clarke, "Optimization and Nonsmooth Analysis," Wiley-Interscience, New York, 1983; Republished as vol. 5 of "Classics in Applied Mathematics," SIAM, 1990.

[3]

F. H. Clarke, Necessary conditions in dynamic optimization, Memoirs of the Amer. Math. Soc., 173 (2005), x+113.

[4]

F. H. Clarke, Necessary conditions in optimal control and in the calculus of variations, in "Differential Equations, Chaos and Variational Problems" (ed. V. Staicu), Birkhäuser, (2008), 143-156. doi: doi:10.1007/978-3-7643-8482-1_11.

[5]

F. H. Clarke and M. R. de Pinho, The nonsmooth maximum principle,, Control and Cybernetics, (). 

[6]

F. H. Clarke and M. R. de Pinho, Optimal control problems with mixed constraints,, SIAM J. Control Optim., (). 

[7]

F. H. Clarke, Yu. Ledyaev and M. R. de Pinho, An extension of the Schwarzkopf multiplier rule in optimal control, preprint, 2009.

[8]

F. H. Clarke and Yu. S. Ledyaev, Mean value inequalities in Hilbert space, Trans. Amer. Math. Soc., 344 (1994), 307-324. doi: doi:10.2307/2154718.

[9]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory," Graduate Texts in Mathematics, vol. 178, Springer-Verlag, New York, 1998.

[10]

M. R. Hestenes, "Calculus of Variations and Optimal Control Theory," Wiley, New York, 1966.

[11]

A. A. Milyutin and N. P. Osmolovskii, "Calculus of Variations and Optimal Control," American Math. Soc., Providence, 1998.

[12]

L. W. Neustadt, "Optimization," Princeton University Press, Princeton, 1976.

[13]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mischenko, "The Mathematical Theory of Optimal Processes," Wiley-Interscience, New York, 1962.

[14]

R. B. Vinter, "Optimal Control," Birkhäuser, Boston, 2000.

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